Derive the insect model and restore its zero equilibrium

An insect population $N$ after $t$ weeks satisfies $dN/dt=N-kN^2$, $k>0$. Show, rather than merely verify, the general solution $N=1/(k+Ae^{-t})$.4 marks

Initially there are $250$ insects and the population tends to $10000$ after a very long time. Find the time to reach three times the initial population, to the nearest day.3 marks